MA-C2 Differential calculus Y12



Year 12 Mathematics AdvancedMA-C2 Differential calculusUnit durationThe topic Calculus involves the study of how things change and provides a framework for developing quantitative models of change and deducing their consequences. It involves the development of two key aspects of calculus, namely differentiation and integration. The study of calculus is important in developing students’ capacity to operate with and model situations involving change, using algebraic and graphical techniques to describe and solve problems and to predict outcomes in fields such as biomathematics, economics, engineering and the construction industry.4 weeksSubtopic focusOutcomesThe principal focus of this subtopic is to develop and apply rules for differentiation to a variety of functions. Students develop an understanding of the interconnectedness of topics from across the syllabus and the use of calculus to help solve problems from each topic. These skills are then applied in the following subtopic on the second derivative in order to investigate applications of the calculus of trigonometric, exponential and logarithmic functions.A student:applies calculus techniques to model and solve problems MA12-3applies appropriate differentiation methods to solve problems MA12-6chooses and uses appropriate technology effectively in a range of contexts, models and applies critical thinking to recognise appropriate times for such use MA12-9constructs arguments to prove and justify results and provides reasoning to support conclusions which are appropriate to the context MA12-10Prerequisite knowledgeAssessment strategiesThe material in this topic builds on content from the Year 11 topics of MA-C1 Introduction to differentiation and MA-E1 Exponential and logarithmic functions. It would also be useful to have completed the Year 12 topic of MA-T3 Trigonometric functions and graphs.Formative assessment: Students to use both online graphing tools and pen-and-paper methods to demonstrate informal and, where appropriate, formal investigations and proofs for the concepts explored in this topic. All outcomes referred to in this unit come from Mathematics Advanced Syllabus? NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2017Glossary of termsTermDescriptionexponential growth and decayExponential growth occurs when the rate of change of a mathematical function is positive and proportional to the function’s current value. Exponential decay occurs in the same way when the growth rate is negative.limitThe limit of a function at a point a, if it exists, is the value the function approaches as the independent variable approaches a.The notation used is: limx→afx=LThis is read as ‘the limit of f(x) as x approaches a is L’.Lesson sequenceContentSuggested teaching strategies and resources Date and initialComments, feedback, additional resources usedEstablishing the derivatives for sin x and cos x (1 or 2 lessons)C2.1: Differentiation of trigonometric, exponential and logarithmic functionsestablish the formulae ddxsinx=cosx and ddxcosx=-sinx by numerical estimations of the limits and informal proofs based on geometric constructions (ACMMM102) Investigating the limits of limx→0sin xx Staff need to determine early that all calculations involving trigonometric functions in calculus are performed using radians and will not be accurate if degrees are used.Students need to establish the limit, limx→0sin xx which represents the gradient of sinx at x=0.Students need to calculate the expressions, in radians, for values of x that approach 0, for example, the values of sinxx when x=0.1, 0.01, 0.001 and 0.0001.Leading to the numerical estimation of limx→0sin xx=1 and an estimate for the gradient of the curve y=sin x at the point x =0A similar investigation can be performed forlimx→0tan xx=1 as both sin x, tan x and x converge at the origin creating an unusual anomaly which can be approximated to the calculation of rmal construction of the gradient of y=sin(x)From the curve of y=sin(x) and using the limit established above, establish the gradient of the curve at points where it can be determinedLeading to the informal results:when x=0, ddxsinx=1 when x=π2, ddxsinx=0 when x=π, ddxsinx=-1 when x=3π2, ddxsinx=0 when x=2π, ddxsinx=1 Sketch these points and lead students to the result for ddx(sinx)=cosxA similar investigation can be structured to investigate the gradient of y=cos xStaff may like to use the formal method of differentiation from first principles to establish the results above but this is not a syllabus requirement. Staff may like to use this Geogebra app to investigate the gradient of the curve y=sin(x) Finding derivatives of expressions involving sin x and cos x(2 or 3 lessons)C2.1: Differentiation of trigonometric, exponential and logarithmic functionscalculate derivatives of trigonometric functionsC2.2: Rules of differentiationapply the product, quotient and chain rules to differentiate functions of the form fxgx, fxgx and f(g(x)) where f(x) and g(x) are any of the functions covered in the scope of this syllabus, for example xex, tanx, 1xn, xsinx, e-xsinx and fax+b (ACMMM106) use the composite function rule (chain rule) to establish and use the derivatives of sin(fx), cos(fx) and tan?(f(x))Finding derivatives of expressions involving sin x and cos xStaff need to deliver questions of the formddx(ksin x) or ddx(kcos x) where k is constantddx(sin kx) or ddx(cos kx) where k is constantExpressions involving a mixture of these terms, polynomial terms and exponential terms.And expressions that require differentiation using the product, quotient and chain rules.Students need to explore derivatives in the form ddx(sinf(x)) where f(x) is a function of x by applying the chain rule. Lead students to the generalised result ddx(sinf(x))=f'(x)cos?f(x). Students need to run a similar investigation into ddx(cosf(x))=-f'(x)sin?f(x)Lead students to use the quotient rule to determine the derivative ddx(tanx)=sec2x by using the identity tanx=sinxcosx.Use this result to lead students to apply the chain rule to determine ddx(tanfx)=f'xsec2f(x)Differentiating exponential functions(2 or 3 lessons)C2.1: Differentiation of trigonometric, exponential and logarithmic functionsestablish and use the formula ddxax= (lna)axusing graphing software or otherwise, sketch and explore the gradient function for a given exponential function, recognise it as another exponential function and hence determine the relationship between exponential functions and their derivatives C2.2: Rules of differentiationapply the product, quotient and chain rules to differentiate functions of the form fxgx, fxgx and f(g(x)) where f(x) and g(x) are any of the functions covered in the scope of this syllabus, for example xex, tanx, 1xn, xsinx, e-xsinx and fax+b (ACMMM106) use the composite function rule (chain rule) to establish that ddxefx=f'(x)ef(x)Differentiating exponential functionsLead students to apply the chain rule to determine ddxefx=f'(x)Review and establish the result eln(a)=a to be used within the proof below.Structure a formal proof by starting with the LHS of the identityLHS=ddxax=ddxelnax=ddxexlna=lnaexlna=lnaax=RHS∴ddxax= lnaaxNote, if y=ax then dydx can be expressed as lna ×yStudents need to answer questions in the formddx2xddx0.4xddx(23)xddx12xddx43xand expressions that include exponential terms as part of product, quotient and chain rule questions.Students need to extend their differentiation skills developed and apply them to find the equation of tangents and normal at various points to the curve.Staff and students may like to use this Geogebra app to demonstrate curves of the function y=ax and its derivative function.Staff may like to reference the Exponential and Logarithmic Functions (PDF) resource published by the Australian Mathematical Sciences Institute. Page 16 refers to this concept.Finding the derivative of logarithmic functions(2 or 3 lessons)C2.1: Differentiation of trigonometric, exponential and logarithmic functionscalculate the derivative of the natural logarithm function ddxlnx=1xestablish and use the formula ddxlogax= 1x lnaC2.2: Rules of differentiationapply the product, quotient and chain rules to differentiate functions of the form fxgx, fxgx and f(g(x)) where f(x) and g(x) are any of the functions covered in the scope of this syllabus, for example xex, tanx, 1xn, xsinx, e-xsinx and fax+b (ACMMM106) use the composite function rule (chain rule) to establish that ddxlnf(x)=f'(x)f(x)use the logarithmic laws to simplify an expression before differentiatingFinding the derivative of logarithmic functionsStaff may like to lead students to the result ddxlnx=1xLet y=lnx∴x=eydxdy=eydydx=1ey∴dydx=1xorddxlnx=1xStaff need to establish the result ddxlogax= 1x lnaProofLHS=ddxlogax=ddxlnxlna using the change of base logarithmic law= 1x lna=RHS∴ddxlogax= 1x lnaStudents need to answer questions in the formddx5lnxddxln3xddxlnx2 [How many ways can the solution be generated? Using the chain rule? Or Logarithmic laws?]ddxlnx+5ddxlog2xddxlog10x3ddx(log3x4)ddx1log5xAnd expressions that include logarithmic terms as part of chain, product and quotient rule questions.Students need to explore differentiation questions of the form ddxlnf(x) where f(x) is some function of x. By analysing the results lead them to develop the generalised expression ddxlnf(x)=f'(x)f(x).Students need to extend their differentiation skills developed and apply it to find the equation of tangents and normal at various points to the curve.Reflection and evaluationPlease include feedback about the engagement of the students and the difficulty of the content included in this section. You may also refer to the sequencing of the lessons and the placement of the topic within the scope and sequence. All ICT, literacy, numeracy and group activities should be recorded in the ‘Comments, feedback, additional resources used’ section. ................
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